Guess and Check Heuristic withTables anf Bar Models

 Number Talk & Thinking Classroom Lesson Plan: Guess and Check Heuristic Using Tables 

 

This Number Talk and Thinking Classroom approach is designed for 4th-6th graders and also serves as a PD (Professional Development) session to introduce teachers to the power of:

• Guess and Check  "heuristics" tables
• Mathematical bar models
• Visual problem-solving using heuristics
• The Concrete-Pictorial-Abstract (CPA) approach

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Guess and Check Heuristic Problem Solving with Bar Models FLIPPED PD PODCAST 

Lesson Flow for Students & PD Session Structure

1️⃣ Warm-Up: Quick Number Talk (5-10 minutes)

Objective: Activate prior knowledge and encourage mental math reasoning.

Prompt:
"If you bought a combination of two items, one costing $7 and another costing $4, and spent exactly $141, how might you figure out how many of each item you bought?"

Facilitation:

• Let students think independently for 1-2 minutes.
• Have them share strategies without solving it completely.
• Encourage multiple representations (e.g., breaking down numbers, estimation, or mental division).

Key Discussion Questions:

• How can you estimate a reasonable starting guess?
• If your guess is too high or too low, what should you do?
• How can a table or a visual math model help organize thinking?

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2️⃣ Hands-On: Thinking Classroom Problem Solving (20-30 minutes)

Students work in groups using vertical whiteboards (or large chart paper) to solve the problem.

Step 1: Present the Word Problem Visually

Write or display:

💭 A box of caramel candy costs $7. A bag of lollipops costs $4. Dad bought 27 items in total and paid $141. How many bags of lollipops did he buy?

Ask:

• What do we know?
• What do we need to find?
• What might be a good first guess?

Step 2: Model with a Bar Diagram (Pictorial Stage)

Guide students to draw a bar model:

• One bar for boxes of candy ($7 per unit).
• One bar for bags of lollipops ($4 per unit).
• A total length of 27 items and a total cost of $141.

Encourage discussion:

• How does the model help visualize the problem?
• What if we start by assuming half-and-half (13 each)?
• What happens if we adjust the numbers?

Step 3: Use a Guess & Check Table

Introduce a table to organize their thinking:

Bags of Lollipops	Boxes of Candy	Lollipops (×$4)	Candy (× $7)	Total Cost
10	17	10 × $4 = $40	17 × $7 = $119	$159 (Too much)
5	22	5 × $4 = $20	22 × $7 = $154	$174 (Too much)
15	12	15 × $4 = $60	12 × $7 = $84	$144 (Too much)
11	16	11 × $4 = $	16 × $7 = $___	$141 ✅ (Correct Answer)

💡 Ask students: What patterns do you notice in the numbers?
💡 How does changing one variable affect the total?

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3️⃣ Student Reflection & Discussion (10 minutes)

Whole-class discussion:

• How did the bar model help you think about the problem?
• How did the Guess & Check table organize your work?
• Why is visualization an important mathematical strategy?
• How could this method apply to real-world problems?

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Professional Development (PD) Adaptation for Teachers

Focus:

• Demonstrate how heuristics like Guess & Check and bar models enhance problem-solving.
• Show the power of pictorial models (CPA) in deepening conceptual understanding.
• Highlight how vertical non-permanent surfaces (Thinking Classroom strategy) engage students in reasoning.

Interactive PD Activity:

1. Teachers solve the problem first, using bar models and Guess & Check.
2. Group discussion on why these heuristics work across grade levels.
3. Debrief on instructional strategies, emphasizing student-led discovery over direct instruction.

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Final Takeaways for Teachers & Students

✅ Bar models make abstract problems visual.
✅ Guess & Check tables encourage systematic problem-solving.
✅ Thinking Classroom strategies (whiteboards, group work) engage students deeply.
✅ Mathematics is about reasoning, not just getting answers.

Please make any suggestions for refinements or additional resources for this PD session? 🚀

To solve this problem using the Guess and Check heuristic from Singapore Math, students would follow a structured approach to systematically test different possible solutions. Here's how it would be done using a Guess and Check table:

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Step 1: Understand the Problem

We are given:

• Each box of caramel candy costs $7.00.
• Each bag of lollipops costs $4.00.
• Dad bought a total of 27 items (boxes + bags).
• Dad spent a total of $141.00.

The goal is to determine how many bags of lollipops Dad bought.

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Step 2: Set Up the Guess and Check Table

A Guess and Check table allows us to systematically try different values for the number of lollipops and adjust based on whether the total cost is too high or too low.

Guess (Bags of Lollipops)	Boxes of Caramel Candies	Cost of Lollipops (× $4)	Cost of Candies (× $7)	Total Cost
10	17	10 × $4 = $40	17 × $7 = $119	$40 + $119 = $159 (Too high)
5	22	5 × $4 = $20	22 × $7 = $154	$20 + $154 = $174 (Too high)
15	12	15 × $4 = $60	12 × $7 = $84	$60 + $84 = $144 (Too high)
6	21	6 × $4 = $24	21 × $7 = $147	$24 + $147 = $171 (Too high)
9	18	9 × $4 = $36	18 × $7 = $126	$36 + $126 = $162 (Too high)
12	15	12 × $4 = $48	15 × $7 = $105	$48 + $105 = $153 (Too high)
8	19	8 × $4 = $32	19 × $7 = $133	$32 + $133 = $165 (Too high)
6	21	6 × $4 = $24	21 × $7 = $147	$24 + $147 = $171 (Too high)
16	11	16 × $4 = $___	11 × $7 = $___	$___ + $___ = $141 (Correct Answer)

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Step 3: Verify the Answer

From the table, we see that when Dad buys ____ bags of lollipops and ____ boxes of caramel candy, the total cost is exactly $141, which matches the given information.

Thus, the correct answer is: Dad bought _____ bags of lollipops.

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How Students Benefit from This Approach

1. Organized Thinking – The table helps students keep track of their guesses and the calculations.
2. Error Checking – They can immediately see if their guess is too high or too low and adjust accordingly.
3. Pattern Recognition – Students start noticing trends in how increasing or decreasing one variable affects the total cost.
4. Logical Deduction – Instead of randomly guessing, students make more informed choices based on previous results.

This method aligns perfectly with Singapore Math’s heuristic-based problem-solving and ensures students develop strong number sense and reasoning skills! 🚀

Alternative Word Problem & Solution

Problem Statement (Edited for Clarity):

A box of candy costs $7.00, and a bag of lollipops costs $4.00. Dad bought a total of 27 items (candies and lollipops combined) and paid $141.00 in total. How many bags of lollipops did Dad buy?

Solution Using the Guess & Check Heuristic:

We set up a table to systematically test different values for the number of lollipops and candies.

You're absolutely right! Let's carefully go through the correct calculation and then provide a revised solution.

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10 More Word Problems for Students to Solve Using Guess & Check Tables

1️⃣ A bookstore sells notebooks for $3 each and pens for $2 each. A teacher buys 25 items for $62. How many notebooks did the teacher buy?

2️⃣ Tickets for a school event cost $8 for adults and $5 for students. If 30 people attended and the total revenue was $190, how many students bought tickets?

3️⃣ A pet store sells goldfish for $6 each and hamsters for $9 each. A customer buys 20 animals and spends $138. How many goldfish did they buy?

4️⃣ A farmer sells apples for $2 per pound and oranges for $3 per pound. He sells 50 pounds of fruit for a total of $120. How many pounds of apples did he sell?

5️⃣ A bakery sells cupcakes for $2 each and pies for $10 each. If a customer buys 15 items and spends $80, how many pies did they buy?

6️⃣ A game store sells board games for $12 each and video games for $25 each. If 20 games were sold for a total of $340, how many board games were sold?

7️⃣ A clothing store sells T-shirts for $8 each and hoodies for $25 each. If 35 items were sold for a total of $410, how many hoodies were sold?

8️⃣ A theater charges $12 for adult tickets and $7 for children’s tickets. If 40 tickets were sold for a total of $370, how many adult tickets were sold?

9️⃣ A music store sells guitars for $150 and keyboards for $100. If 12 instruments were sold for a total of $1,500, how many guitars were sold?

🔟 A bike shop sells helmets for $20 each and bicycles for $150 each. If 15 items were sold for a total of $1,000, how many helmets were sold?

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Teaching Tip: Encourage students to create Guess & Check tables for each problem, adjusting their guesses systematically to find the correct answer. This will strengthen their problem-solving skills, numerical reasoning, and logical thinking. 🚀

Let's carefully go through the solution step by step and verify the correct answer.

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Step 1: Define the Variables

• Let x = number of lollipops.
• Let y = number of boxes of candy.

We have two equations based on the problem:

1. Total items equation: $$x + y = 27$$
2. Total cost equation: $$4x + 7y = 141$$

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Step 2: Solve the System of Equations

Step 2.1: Express $y$ in terms of $x$

From equation (1):

$$y = 27 - x$$

Step 2.2: Substitute into equation (2)

$$4x + 7(27 - x) = 141$$

Expand:

$$4x + 189 - 7x = 141$$

Combine like terms:

$$-3x + 189 = 141$$

Subtract 189 from both sides:

$$-3x = -48$$

Divide by -3:

$$x = 16$$

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Step 3: Find $y$

$$y = 27 - 16 = 11$$

So, Dad bought 16 bags of lollipops and 11 boxes of candy. ✅

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Step 4: Verify the Solution

1. Total items check:

   $$16 + 11 = 27 \quad \text{✅ Correct}$$

2. Total cost check:

   $$(16 \times 4) + (11 \times 7) = 64 + 77 = 141 \quad \text{✅ Correct}$$

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Final Answer:

✅ Dad bought 16 bags of lollipops and 11 boxes of candy. 🎯

Thanks for your patience! You were right—16 is the correct answer. 🚀

I'll solve this problem both algebraically and with a guess and check approach.

Let's start by defining our variables:

• Let x = number of $4 items
• Let y = number of $7 items

We know that the total cost is $141, so we can write: 4x + 7y = 141

Algebraic Solution:

I'll solve for x in terms of y: 4x = 141 - 7y x = (141 - 7y) ÷ 4

Since x must be a whole number (we can't buy a fraction of an item), we need (141 - 7y) to be divisible by 4.

Let's test different values of y:

• If y = 1: x = (141 - 7) ÷ 4 = 134 ÷ 4 = 33.5 (not a whole number)
• If y = 2: x = (141 - 14) ÷ 4 = 127 ÷ 4 = 31.75 (not a whole number)
• If y = 3: x = (141 - 21) ÷ 4 = 120 ÷ 4 = 30 (whole number)

So one solution is x = 30 and y = 3.

Let's continue checking:

• If y = 7: x = (141 - 49) ÷ 4 = 92 ÷ 4 = 23 (whole number)
• If y = 11: x = (141 - 77) ÷ 4 = 64 ÷ 4 = 16 (whole number)
• If y = 15: x = (141 - 105) ÷ 4 = 36 ÷ 4 = 9 (whole number)
• If y = 19: x = (141 - 133) ÷ 4 = 8 ÷ 4 = 2 (whole number)
• If y = 20: x = (141 - 140) ÷ 4 = 1 ÷ 4 = 0.25 (not a whole number)

Guess and Check Table:

Number of $7 items (y)	Cost of $7 items	Number of $4 items (x)	Cost of $4 items	Total Cost
3	$21	30	$120	$141
7	$49	23	$92	$141
11	$77	16	$64	$141
15	$105	9	$36	$141
19	$133	2	$8	$141

The solutions are:

1. 3 items at $7 and 30 items at $4
2. 7 items at $7 and 23 items at $4
3. 11 items at $7 and 16 items at $4
4. 15 items at $7 and 9 items at $4
5. 19 items at $7 and 2 items at $4

Each of these combinations will give you a total cost of exactly $141.